So, that's the question. I dare to generalize it even wider: what branch of mathematics improves the general thinking ability, intilligence, the way the person thinks, and makes it more logical? I'm anticipating answers like "every branch", but I guess there are ones that do it better.

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There really is no "correct answer", because most highly intellectual endeavors improve one's capacity to think, and enhance logical thought. But there are different "ways of thinking" that are all valuable, and which "ways of thinking" that a branch or subject engages and develops depends on the topic/branch.

But in general, there is no replacement for courses that emphasize critical thinking, problem solving, and introductory logic ("Elementary Logic", "Symbolic Logic") the latter sometimes offered in math departments, but also at times by Philosophy departments. Course offerings such as Discrete Mathematics cover introductory logic, elementary set theory, and a flexible array of topics which require problem solving of one form or another. For those planning to major in mathematics, there are often classes that provide a more formal introduction to proofs and higher-level mathematics, which are valuable even to those not planning to major in math.

If you are sincerely interested in improving the clarity and quality of your thinking, and becoming a versatile thinker, particularly within mathematics, I recommend each and every of the following references: